$$\log_{b}(M \times N) = \log_{b}(M) \times \log_{b}(N)$$

Question 1

Which of the following correctly applies the Product Rule to expand $$\log_{3}(12x)$$?

Accepted Answer

$$\log_{3}(12) + \log_{3}(x)$$

Question 2

Expand $$\log_{5}\left(\frac{7x^2$}{y^3$}\right)$$ using logarithm properties.

Accepted Answer

$$\log_{5}(7) + 2\log_{5}(x) - 3\log_{5}(y)$$

Question 3

Condense $$2\ln(x) + \frac{1}{2}\ln(y)$$ into a single logarithm.

Accepted Answer

$$\ln(x^2$ \sqrt{y})$$

Question 4

Which of the following is a common mistake when expanding logarithms?

Accepted Answer

$$\log_{b}(M + N) = \log_{b}(M) + \log_{b}(N)$$

Question 5

Expand $$\log_{2}(x^3$ \sqrt{y})$$ using logarithm properties.

Accepted Answer

$$3\log_{2}(x) + \frac{1}{2}\log_{2}(y)$$

Question 6

Condense $$\log_{7}(a) - 2\log_{7}(b) + \frac{1}{3}\log_{7}(c)$$ into a single logarithm.

Accepted Answer

$$\log_{7}\left(\frac{a c$$^{1/3}$$}{b^2$}\right)$$

Question 7

Using the change-of-base formula, how would you rewrite $$\log_{4}(20) in $$terms of natural logarithms?

Accepted Answer

$$\frac{\ln(20)}{\ln(4)}$$

Question 8

A scientist needs to expand $$\log_{10}(x^2$ y^5$)$$ for a calculation. Which is the correct expansion?

Accepted Answer

$$2\log_{10}(x) + 5\log_{10}(y)$$

Question 9

Condense $$\log_{3}(7) + \log_{3}(x) - \log_{3}(y^2$)$$ into a single logarithm.

Accepted Answer

$$\log_{3}\left(\frac{7x}{y^2$}\right)$$

Question 10

Expand $$\log_{b}\left(\frac{\sqrt{x}}{y^4$}\right)$$ using logarithm properties.

Accepted Answer

$$\frac{1}{2}\log_{b}(x) - 4\log_{b}(y)$$

Question 11

Which of the following is NOT a valid logarithm property?

Accepted Answer

$$\log_{b}(M 	imes N) = \log_{b}(M) 	imes \log_{b}(N)$$

Question 12

A medical researcher condenses $$\ln(a) + 2\ln(b) - \ln(c)$$ for a formula. What is the result?

Accepted Answer

$$\ln\left(\frac{a b^2$}{c}\right)$$

Question 13

Expand $$\log_{2}(x^4$ y$$^{1/3}$$)$$ using logarithm properties.

Accepted Answer

$$4\log_{2}(x) + \frac{1}{3}\log_{2}(y)$$

Question 14

Condense $$\frac{1}{4}\log_{b}(x) - 2\log_{b}(5) - 10\log_{b}(y)$$ into a single logarithm.

Accepted Answer

$$\log_{b}\left(\frac{\sqrt[4]{x}}{25 y$$^{10}$$}\right)$$

Question 15

Which of the following is the correct change-of-base formula for $$\log_{b}(M)$$?

Accepted Answer

$$\frac{\log_{a}(M)}{\log_{a}(b)}$$

Question 16

Expand $$\log_{6}(3^9$)$$ using the Power Rule.

Accepted Answer

$$9\log_{6}(3)$$

Question 17

A chemist condenses $$\log_{10}(50) + \log_{10}(2)$$ for a calculation. What is the result?

Accepted Answer

$$\log_{10}(100)$$

Question 18

Expand $$\ln\left(\frac{e^5$}{11}\right)$$ using logarithm properties.

Accepted Answer

$$5 - \ln(11)$$

Question 19

Condense $$2\log(x - 3) - \log(x)$$ into a single logarithm.

Accepted Answer

$$\log\left(\frac{(x - 3)^2$}{x}\right)$$

Properties of Logarithms: Product, Quotient, Power, Expansion, Condensing, and Change-of-Base

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